Problem: All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$3.00$ each for students, and the group paid $$51.00$ in total. The next month, the same group visited a science museum where the tickets cost $$30.00$ each for teachers and $$10.50$ each for students, and the group paid $$193.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3y = 51}$ ${30x+10.5y = 193.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-12y = -204}$ ${30x+10.5y = 193.5}$ Add the top and bottom equations together. $ -1.5y = -10.5 $ $ y = \dfrac{-10.5}{-1.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7.5x+3y = 51}$ to find $x$ ${7.5x + 3}{(7)}{= 51}$ $7.5x+21 = 51$ $7.5x = 30$ $x = \dfrac{30}{7.5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {30x+10.5y = 193.5}$ and get the same answer for $x$ ${30x + 10.5}{(7)}{= 193.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.